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偏微分方程式の有限要素解の最大値ノルム誤差評価
http://hdl.handle.net/10191/0002000948
http://hdl.handle.net/10191/000200094803753a2e-869d-46be-b818-f6e05efe1bca
| 名前 / ファイル | ライセンス | アクション |
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本文 (749KB)
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要旨 (302KB)
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| Item type | 学位論文 / Thesis or Dissertation(1) | |||||||
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| 公開日 | 2023-04-24 | |||||||
| タイトル | ||||||||
| タイトル | Maximum norm error estimation for the finite element solution to partial differential equations | |||||||
| 言語 | en | |||||||
| タイトル | ||||||||
| タイトル | 偏微分方程式の有限要素解の最大値ノルム誤差評価 | |||||||
| 言語 | ja | |||||||
| 言語 | ||||||||
| 言語 | eng | |||||||
| キーワード | ||||||||
| 言語 | en | |||||||
| 主題Scheme | Other | |||||||
| 主題 | Lagrange interpolation | |||||||
| キーワード | ||||||||
| 言語 | en | |||||||
| 主題Scheme | Other | |||||||
| 主題 | Fujino-Morley FEM space | |||||||
| キーワード | ||||||||
| 言語 | en | |||||||
| 主題Scheme | Other | |||||||
| 主題 | Bernstein polynomials | |||||||
| キーワード | ||||||||
| 言語 | en | |||||||
| 主題Scheme | Other | |||||||
| 主題 | finite element method | |||||||
| キーワード | ||||||||
| 言語 | en | |||||||
| 主題Scheme | Other | |||||||
| 主題 | boundary value problem | |||||||
| キーワード | ||||||||
| 言語 | en | |||||||
| 主題Scheme | Other | |||||||
| 主題 | maximum norm error estimation | |||||||
| 資源タイプ | ||||||||
| 資源タイプ識別子 | http://purl.org/coar/resource_type/c_db06 | |||||||
| 資源タイプ | doctoral thesis | |||||||
| アクセス権 | ||||||||
| アクセス権 | open access | |||||||
| アクセス権URI | http://purl.org/coar/access_right/c_abf2 | |||||||
| 著者 |
Galindo, Shirley Mae Alfarero
× Galindo, Shirley Mae Alfarero
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| 抄録 | ||||||||
| 内容記述タイプ | Abstract | |||||||
| 内容記述 | The Lagrange interpolation is one of the most used interpolation types to approximate functions. Its interpolation error has been estimated under various norms and is a widely explored topic in numerical analysis. Error estimation for the approximation of functions greatly affects the development of numerical solutions to partial differential equations. For example, the maximum norm of the interpolation error influences the discretization error of the finite element method (FEM) solution. In this research, we consider the L ∞-norm estimation for the linear Lagrange interpolation over a triangle element K by using the H2-seminorm of the objective function, that is, to obtain the explicit estimate of the constant satisfying ∥u−ΠLu∥∞,K ≤ CL(K)|u|2,K, ∀u ∈ H2(K). Here, CL(K) is the interpolation error constant to be evaluated explicitly. Waldron (1998) estimated the maximum norm of the Lagrange interpolation error in terms of the maximum of the second derivative of the objective function, i.e., ∥u(2)∥∞,K. Since the H2-seminorm of the function requires less function regularity than ∥u(2)∥∞,K norm, the result of this research has more application. For triangle element K of general shape, a formula to give an estimate of the interpolation error constant CL(K) is obtained through theoretical analysis. The theoretical estimation leads to a raw bound that works well for triangle of arbitrary shapes. Particularly, our analysis tells that the value of CL(K) can be very large and tend to ∞ if the triangle element tends to degenerate to a 1D segment. An algorithm for the optimal estimation for CL(K) is also proposed, which extends the technique of eigenvalue estimation by Liu [17]. The objective problem is converted to a quadratic minimization problem with maximum norm constraint. To overcome the difficulty caused by the maximum norm in the constraint condition, a novel method is proposed by utilizing the orthogonality property of the interpolation associated to the Fujino-Morley FEM space and the convex-hull property of the Bernstein representation of functions in the FEM space. Specifically, for a unit right isosceles triangle K, it has been shown by rigorous computation that 0.40432 ≤ CL(K) ≤ 0.41596. Also, the maximum norm error estimation of the Lagrange interpolation has been used to estimate the local maximum error of the FEM solution to the Poisson boundary value problem. That is, given the subdomain Ω' ⊆ Ω, an estimator ξ is desired which satisfies ∥u−uh∥∞,Ω' ≤ ξ (u : exact solution, uh : FEM solution). To compute the above estimator, Fujita's method of pointwise estimation of the boundary value solution is employed, as well as the interpolation error estimation for the Lagrange interpolation. The code is shared at https://ganjin.online/shirley/InterpolationErrorEstimate. Main results of this dissertation can be found in [13]. | |||||||
| 言語 | en | |||||||
| 学位名 | ||||||||
| 言語 | ja | |||||||
| 学位名 | 博士（理学） | |||||||
| 学位授与機関 | ||||||||
| 学位授与機関識別子Scheme | kakenhi | |||||||
| 学位授与機関識別子 | 13101 | |||||||
| 言語 | ja | |||||||
| 学位授与機関名 | 新潟大学 | |||||||
| 言語 | en | |||||||
| 学位授与機関名 | Niigata University | |||||||
| 学位授与年月日 | ||||||||
| 学位授与年月日 | 2022-09-20 | |||||||
| 学位授与番号 | ||||||||
| 学位授与番号 | 甲第5092号 | |||||||
| 学位記番号 | ||||||||
| 内容記述タイプ | Other | |||||||
| 内容記述 | 新大院博(理)第479号 | |||||||
| 言語 | ja | |||||||
